def is_rep(self, image, n, force=False):
"""
Returns `True` if the map generated by mapping the generators to the matrices defined in
`image` is an `n`-dimensional representation of `self`, and `False` otherwise. The entries
of `image` must be `n`-by-`n` matrices with entries in the algebraic closure of the base
field of `self`. Its length must match the number of generators of `self.`
Use `force=True` if the function does not recognize the base field as computable, but the
field is computable.
"""
if (not force and self.base_field() not in NumberFields
and self.base_field() not in FiniteFields):
raise TypeError('Base field must be computable. If %s is' %
self.base_field() +
' computable then use force=True to bypass this.')
if n not in ZZ or n < 1:
raise ValueError('Dimension must be a positive integer.')
M = MatrixSpace(self.base_field().algebraic_closure(), n, sparse=True)
if len(image) != self.ngens():
raise ValueError(
'Length of image does not match number of generators.')
if False in {mat in M for mat in image}:
raise TypeError('Improper image, must contain elements of %s.' %
M._repr_())
image = [M(image[i]) for i in range(len(image))]
for rel in self.rels():
if self._to_matrix(rel, M, image) != M.zero(): return False
return True
def to_matrix(self):
r"""
Return ``self`` as a matrix.
We define a matrix `M_{xy} = \alpha(x, y)` for some element
`\alpha \in I_P` in the incidence algebra `I_P` and we order
the elements `x,y \in P` by some linear extension of `P`. This
defines an algebra (iso)morphism; in particular, multiplication
in the incidence algebra goes to matrix multiplication.
EXAMPLES::
sage: P = posets.BooleanLattice(2)
sage: I = P.incidence_algebra(QQ)
sage: I.moebius().to_matrix()
[ 1 -1 -1 1]
[ 0 1 0 -1]
[ 0 0 1 -1]
[ 0 0 0 1]
sage: I.zeta().to_matrix()
[1 1 1 1]
[0 1 0 1]
[0 0 1 1]
[0 0 0 1]
TESTS:
We check that this is an algebra (iso)morphism::
sage: P = posets.BooleanLattice(4)
sage: I = P.incidence_algebra(QQ)
sage: mu = I.moebius()
sage: (mu*mu).to_matrix() == mu.to_matrix() * mu.to_matrix()
True
"""
P = self.parent()
MS = MatrixSpace(P.base_ring(),
P._poset.cardinality(),
sparse=True)
L = P._linear_extension
M = copy(MS.zero())
for i, c in self:
M[L.index(i[0]), L.index(i[1])] = c
M.set_immutable()
return M
def to_matrix(self):
r"""
Return ``self`` as a matrix.
We define a matrix `M_{xy} = \alpha(x, y)` for some element
`\alpha \in I_P` in the incidence algebra `I_P` and we order
the elements `x,y \in P` by some linear extension of `P`. This
defines an algebra (iso)morphism; in particular, multiplication
in the incidence algebra goes to matrix multiplication.
EXAMPLES::
sage: P = posets.BooleanLattice(2)
sage: I = P.incidence_algebra(QQ)
sage: I.moebius().to_matrix()
[ 1 -1 -1 1]
[ 0 1 0 -1]
[ 0 0 1 -1]
[ 0 0 0 1]
sage: I.zeta().to_matrix()
[1 1 1 1]
[0 1 0 1]
[0 0 1 1]
[0 0 0 1]
TESTS:
We check that this is an algebra (iso)morphism::
sage: P = posets.BooleanLattice(4)
sage: I = P.incidence_algebra(QQ)
sage: mu = I.moebius()
sage: (mu*mu).to_matrix() == mu.to_matrix() * mu.to_matrix()
True
"""
P = self.parent()
MS = MatrixSpace(P.base_ring(), P._poset.cardinality(), sparse=True)
L = P._linear_extension
M = copy(MS.zero())
for i, c in self:
M[L.index(i[0]), L.index(i[1])] = c
M.set_immutable()
return M
def __call__(self, A, name=''):
r"""
Create an element of the homspace ``self`` from `A`.
INPUT:
- ``A`` -- one of the following:
- an element of a Hecke algebra
- a Hecke module morphism
- a matrix
- a list of elements of the codomain specifying the images
of the basis elements of the domain.
EXAMPLES::
sage: M = ModularForms(Gamma0(7), 4)
sage: H = M.Hom(M)
sage: H(M.hecke_operator(7))
Hecke module morphism T_7 defined by the matrix
[ -7 0 0]
[ 0 1 240]
[ 0 0 343]
Domain: Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) ...
Codomain: Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) ...
sage: H(H(M.hecke_operator(7)))
Hecke module morphism T_7 defined by the matrix
[ -7 0 0]
[ 0 1 240]
[ 0 0 343]
Domain: Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) ...
Codomain: Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) ...
sage: H(matrix(QQ, 3, srange(9)))
Hecke module morphism defined by the matrix
[0 1 2]
[3 4 5]
[6 7 8]
Domain: Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) ...
Codomain: Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) ...
TESTS:
Make sure that the element is created correctly when the codomain is
not the full module (related to :trac:`21497`)::
sage: M = ModularSymbols(Gamma0(3),weight=22,sign=1)
sage: S = M.cuspidal_subspace()
sage: H = S.Hom(S)
sage: H(S.gens())
Hecke module morphism defined by the matrix
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
Domain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...
sage: H.zero() in H
True
sage: H.one() in H
True
"""
try:
if A.parent() == self:
A._set_parent(self)
return A
A = A.hecke_module_morphism()
if A.parent() == self:
A._set_parent(self)
return A
else:
raise TypeError("unable to coerce A to self")
except AttributeError:
pass
if A in self.base_ring():
dim_dom = self.domain().rank()
dim_codom = self.codomain().rank()
MS = MatrixSpace(self.base_ring(), dim_dom, dim_codom)
if self.domain() == self.codomain():
A = A * MS.identity_matrix()
elif A == 0:
A = MS.zero()
else:
raise ValueError('scalars do not coerce to this homspace')
elif isinstance(A, (list, tuple)):
A = matrix([self.codomain().coordinate_vector(f) for f in A])
return HeckeModuleMorphism_matrix(self, A, name)
